Given's Rotation SVD example

Average Error: 13.0 → 13.7

Time: 6.4s

Precision: 64

\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]

Math

\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]

↓

\[\sqrt{0.5 \cdot \left(1 + \frac{x}{{\left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}^{\frac{3}{4}} \cdot \sqrt{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]

C

double f(double p, double x) {
        double r258889 = 0.5;
        double r258890 = 1.0;
        double r258891 = x;
        double r258892 = 4.0;
        double r258893 = p;
        double r258894 = r258892 * r258893;
        double r258895 = r258894 * r258893;
        double r258896 = r258891 * r258891;
        double r258897 = r258895 + r258896;
        double r258898 = sqrt(r258897);
        double r258899 = r258891 / r258898;
        double r258900 = r258890 + r258899;
        double r258901 = r258889 * r258900;
        double r258902 = sqrt(r258901);
        return r258902;
}

↓

double f(double p, double x) {
        double r258903 = 0.5;
        double r258904 = 1.0;
        double r258905 = x;
        double r258906 = 4.0;
        double r258907 = p;
        double r258908 = r258906 * r258907;
        double r258909 = r258908 * r258907;
        double r258910 = r258905 * r258905;
        double r258911 = r258909 + r258910;
        double r258912 = sqrt(r258911);
        double r258913 = 0.75;
        double r258914 = pow(r258912, r258913);
        double r258915 = sqrt(r258912);
        double r258916 = sqrt(r258915);
        double r258917 = r258914 * r258916;
        double r258918 = r258905 / r258917;
        double r258919 = r258904 + r258918;
        double r258920 = r258903 * r258919;
        double r258921 = sqrt(r258920);
        return r258921;
}

Error

Bits error versus p

Bits error versus x

Target

Original	13.0
Target	13.0
Herbie	13.7

\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

1. Initial program 13.0

   \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
2. Using strategy rm

3. Applied add-sqr-sqrt 13.0

   \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]

4. Applied sqrt-prod 14.2

   \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]
5. Using strategy rm

6. Applied add-sqr-sqrt 14.2

   \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt{\color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)}\]

7. Applied sqrt-prod 14.2

   \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\color{blue}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)}\]

8. Applied sqrt-prod 14.5

   \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \color{blue}{\left(\sqrt{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}\right)}\]

9. Applied associate-*r* 14.3

   \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{\left(\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) \cdot \sqrt{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)}\]

10. Simplified 14.7

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{{\left(\sqrt{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}^{3}} \cdot \sqrt{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]
11. Using strategy rm

12. Applied pow1/2 14.7

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{{\left(\sqrt{\color{blue}{{\left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}^{\frac{1}{2}}}}\right)}^{3} \cdot \sqrt{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]

13. Applied sqrt-pow1 14.6

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{{\color{blue}{\left({\left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}}^{3} \cdot \sqrt{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]

14. Applied pow-pow 13.7

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{{\left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}^{\left(\frac{\frac{1}{2}}{2} \cdot 3\right)}} \cdot \sqrt{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]

15. Simplified 13.7

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{{\left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}^{\color{blue}{\frac{3}{4}}} \cdot \sqrt{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]

16. Final simplification 13.7

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{{\left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}^{\frac{3}{4}} \cdot \sqrt{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1 (/ (* 2 p) x)))))

  (sqrt (* 0.5 (+ 1 (/ x (sqrt (+ (* (* 4 p) p) (* x x))))))))